"We consider certain distinguished extensions of the field F_p((t)) of formal Laurent series over F_p, and look at questions about their model theory and Galois theory, with a particular focus on decidability."

Algebraically closed fields, and in general varieties are among the first examplesof Zariski Geometries.I will consider specialisations of algebraically closed fields and varieties.In the case of an algebraically closed field K, I will show that a specialisationis essentially a residue map, res from K to a residue field k.In both cases I will show universality of the specialisation is controlled by thetranscendence degree of K over k.

Given a field K and an ordered abelian group G, we can form the field K((G)) of generalised formal power series with coefficients in K and indices in G. When is this field decidable? In certain cases, decidability reduces to that of K and G. We survey some results in the area, particularly in the case char K > 0, where much is still unknown.